Strong convergence of adaptive time-stepping schemes for the stochastic Allen--Cahn equation

TL;DR

This paper introduces adaptive time-stepping schemes for the stochastic Allen--Cahn equation, ensuring strong convergence and improved computational efficiency over traditional uniform timestep methods.

Contribution

It proposes and analyzes adaptive schemes that prevent divergence and establish convergence with explicit order, improving numerical stability and efficiency.

Findings

Adaptive schemes converge strongly with order β/2 in time.

They achieve convergence order β/d in space for dimensions 1 to 3.

Numerical experiments confirm lower computational cost and simplicity.

Abstract

It is known in \cite{beccari} that the standard explicit Euler-type scheme (such as the exponential Euler and the linear-implicit Euler schemes) with a uniform timestep, though computationally efficient, may diverge for the stochastic Allen--Cahn equation. To overcome the divergence, this paper proposes and analyzes adaptive time-stepping schemes, which adapt the timestep at each iteration to control numerical solutions from instability. The \textit{a priori} estimates in $\mathcal {C}(\mathcal {O})$-norm and $\dot{H}^{\beta}(\mathcal{O})$-norm of numerical solutions are established provided the adaptive timestep function is suitably bounded, which plays a key role in the convergence analysis. We show that the adaptive time-stepping schemes converge strongly with order $\frac{\beta}{2}$ in time and $\frac{\beta}{d}$ in space with $d$ ($d=1,2,3$) being the dimension and $\beta\in(0,2]$. Numerical experiments show that the adaptive time-stepping schemes are simple to implement and at a lower computational cost than a scheme with the uniform timestep.